The recent discovery of high-temperature superconductivity in multilayer nickelates has raised fundamental questions about its electronic origins and possible connection to the cuprates. Here, we identify a common electronic phenomenology across multilayer nickelates, including signatures of a doping-dependent incommensurate spin-density wave instability coherent enough to reconstruct and partially gap the Fermi surface. We achieve this by exploiting the natural polymorphism between bilayer (2222) and alternating monolayer-trilayer (1313) stacking sequences in bulk La3Ni2O7 crystals, and combining angle-resolved photoemission spectroscopy (ARPES) with effective tight-binding modelling. Polarization-dependent ARPES further reveals that the first electron-removal states are dominated by oxygen-centred planar orbitals: along the Fermi surface, their symmetry evolves from the d3x2−r2 and d3y2−r2 character of 3-spin polarons (3SP) to the familiar dx2−y2 character of Zhang-Rice singlets (ZRS) in cuprates. Doping, and thus the occupation of these orbitals, controls the Fermi-surface topology and the competition between magnetic and superconducting instabilities. Our results establish a direct correspondence between the low-energy electronic structure of layered nickelates and cuprates, pointing to a common microscopic origin of their unconventional superconductivity despite the multi-orbital character of the nickelates.